I don't do TRI, so I may be wide of the mark, but:
Euribor is not an instrument. It originated as a fixing to reflect the cost of borrowing for a term (here 3m) in the interbank market. But it is not an instrument, so there is no return to reinvest, nor an instrument to reinvest in.
Instruments that do depend on 3m Euribor are FRAs, Futures, IRS, and then some less liquid ones like swap-futures, CMS. But they are generally Par instruments, so they will generally net zero (adjusted for collateralisation), unlike a share or other yielding instrument. FRAs don't trade for Spot after the fix, and don't represent a thing that yields a later cashflow after the start date of the FRA. Futures are already a thing. I think the closest thing is a CMS, but that is already pretty much what you'd be making the TRI for.
If you're less wedded to the Euribor fixings and more to the interest rate side, then look at the OIS market - an OIS is intended to reflect a deposit rolled overnight and accrued at the Overnight fixing over a given period; the OIS fixed rate is thus the fair fix based on market expectations of the Overnight fixing for a term.
You could perhaps compound up the Overnight fixing every night and give the result as an index number, but it's fairly pointless since there is already (a) a very liquid market in OIS themselves, (b) a general assumption that the Overnight is the risk free rate, so the index is essentially just tracking the unit in which everything is already denominated and (c) no need to construct another index on something widely understood and modelled.
What are you trying to achieve? Have I missed something?