Interstate 105 was created in 1956 as part of the National System of Interstate and Defense Highways Act of 1956. This was one of the four proposed freeways in and around Eugene that were actually constructed, but I bet it wasn't without its share of problems.
Interstate 105 began construction around this time and was completed between Laura and "Q" Streets (near current Exit 5) and Coberg Rd. (current Exit 2) in 1960, coinciding with Interstate 5's completion between Albany and Eugene. It is unclear if Interstate 105 was marked along the whole route or if the part east of Interstate 5 was umarked (it was not a part of the recently created US-126, as it was in 1972, when US-126 was decommissioned, that the OR-126 designation was applied to the freeway), but currently Interstate 105 is only marked west of I-5. The freeway, then named the "Q" St. Freeway, pressed on past Coberg Rd. to end at Willagillespie Rd. (near current Exit 1). This mile-long segment wouldn't open until 1966 with Lane County's completion of the Delta Highway (unnumbered), and the Washington-Jefferson Street Bridge over the Willamette River towards Central Eugene, originally slated to open in 1966, wouldn't open until 1968; temporary ramps were installed at the south end of the bridge, connecting to W 1st Ave.
In the meantime, Eugene tried to figure out the final design and termination of Interstate 105 while at the same time trying to kill the Roosevelt Freeway, a freeway that was supposed to be the realignment of OR-126 through Eugene. My assumption is that if the Roosevelt would've been built, it would've intersected Interstate 105 just north of W 1st Ave. around current Washington/Jefferson Park (and probably be labeled Exit 1A), and OR-126 wouldn't be cosigned with Interstate 105 south of that interchange. But alas, we need not worry about all these facts, as the Roosevelt was killed, along with Interstates 305 and 505 in Salem and Portland respectively and a slough of other Portland-area freeways. Interstate 105, however, escaped the chopping block, and was eventually completed in Springfield to Main Street (then US-126, now OR-126 Business) in 1971 and in Eugene via an elevated section from W 1st. Ave. to W 7th Ave. in 1973. As mentioned previous, OR-126 was applied to the freeway in full as its new routing through Eugene and Springfield in 1972; this most likely was their original plan when they were also designing the Roosevelt Freeway.
For the next 20 years, the Interstate had an odd sort of exit numbering west of Interstate 5 (I doubt it was numbered east of I-5); the exit numbers and mileposts were reversed. For example, Exit (0) at I-5 was at milepost 4, and Exit 4 in central Eugene was at milepost 0. Not only was this confusing, but it broke the rules of Oregon's milepost numbering and Interstate standards, so in the early 1990's, ODOT renumbered the exits the correct way. Some maps still list the incorrect exit order, Rand McNally ones in particular. While exits east of I-5 aren't numbered, Oregon's trend recently is to number all exits on all state-operated freeways, so I wouldn't be surprised if they will be soon. The interchange with Interstate 5, like all other Interstate termini in Oregon, is not numbered except on one sign heading westbound on OR-126 as the freeway approaches the I-5 North ramp; it is labeled on an exit tab and a gore sign as Exit 4B.
Interestingly enough, on a recent trip down Eugene way, I noticed a construction message sign announcing that the ramp from I-5 South to I-105 East was closed. This was most likely a mistake made by the contractor, but still could indicate ODOT's intentions on possibly extending I-105 into Springfield officially.
EXITS (1A), (1B) & 1(C): These should be the numbers for the OR-99 South/OR-126 Bus., OR-99 North/OR-126 West, and Delta Highway ramps respectively heading west (really south) on I-105 in Eugene. I guess this is consistent with ODOT's refusal to number interstate termini, even though the OR-99 South/OR-126 East ramp isn't the end of I-105...
EXITS (4A) & (4B): The official end of I-105, but not the freeway, so once again, the numbers aren't marked, except for Exit 4B westbound as previously mentioned. One of the gore point signs (I think it's the one for Exit (4B) east to I-5 North) doesn't even say "Exit" with an arrow; it said "[I-5 shield] North" with an arrow, similar to the old junction signs found at some freeway junctions in Southern states and older ones in Pennsylvania.