How can I tune the initial TCP retransmit timeout?
Nope, you can't; it is hardcoded in the kernel. So change the kernel and recompile.
#define TCP_TIMEOUT_INIT ((unsigned)(3*HZ)) /* RFC 1122 initial RTO value */
This is what you should get in your include/net/tcp.h.
But I can see someone provided a patch, even though never tried it myself
The initial setting should not affect your overall performance much, as RTO self-adjusts to network conditions. If you do change RTO, you can set it to 1 sec (but no lower).
There is a discussion of this in RFC 1122:
The following values SHOULD be used to initialize the estimation parameters for a new connection:
(a) RTT = 0 seconds. (b) RTO = 3 seconds. (The smoothed variance is to be initialized to the value that will result in this RTO). The recommended upper and lower bounds on the RTO are known to be inadequate on large internets. The lower bound SHOULD be measured in fractions of a second (to accommodate high speed LANs) and the upper bound should be 2*MSL, i.e., 240 seconds. DISCUSSION: Experience has shown that these initialization values are reasonable, and that in any case the Karn and Jacobson algorithms make TCP behavior reasonably insensitive to the initial parameter choices.
RFC 6298 is a proposed update (published June 2011) that says that RTO can be initialized to a lower value (but no lower than 1 sec), and contains an Appendix containing data that justifies 1 sec as a reasonable initial value.