DAVdroid is using these cipher suites:

D/HttpClient(21404): Enabled protocols: [TLSv1, TLSv1.1, TLSv1.2]

D/HttpClient(21404): Enabled cipher suites:[TLS_ECDH_RSA_WITH_3DES_EDE_CBC_SHA, TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA, TLS_DHE_RSA_WITH_AES_128_CBC_SHA, TLS_ECDH_RSA_WITH_AES_128_CBC_SHA, TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA, SSL_DHE_RSA_EXPORT_WITH_DES40_CBC_SHA, SSL_RSA_EXPORT_WITH_RC4_40_MD5, TLS_DHE_RSA_WITH_AES_256_CBC_SHA, SSL_DHE_RSA_WITH_3DES_EDE_CBC_SHA, TLS_ECDH_RSA_WITH_AES_256_CBC_SHA, SSL_RSA_WITH_RC4_128_SHA, TLS_ECDH_ECDSA_WITH_3DES_EDE_CBC_SHA, SSL_DHE_DSS_EXPORT_WITH_DES40_CBC_SHA, TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA, TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA, SSL_DHE_DSS_WITH_DES_CBC_SHA, TLS_ECDHE_RSA_WITH_RC4_128_SHA, TLS_ECDH_ECDSA_WITH_RC4_128_SHA, TLS_DHE_DSS_WITH_AES_256_CBC_SHA, TLS_RSA_WITH_AES_128_CBC_SHA, SSL_DHE_RSA_WITH_DES_CBC_SHA, TLS_ECDHE_ECDSA_WITH_RC4_128_SHA, TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA, TLS_ECDH_ECDSA_WITH_AES_256_CBC_SHA, TLS_RSA_WITH_AES_256_CBC_SHA, SSL_RSA_WITH_DES_CBC_SHA, TLS_ECDHE_ECDSA_WITH_3DES_EDE_CBC_SHA, TLS_ECDH_RSA_WITH_RC4_128_SHA, TLS_EMPTY_RENEGOTIATION_INFO_SCSV, SSL_RSA_EXPORT_WITH_DES40_CBC_SHA, TLS_ECDHE_RSA_WITH_3DES_EDE_CBC_SHA, SSL_RSA_WITH_RC4_128_MD5, TLS_DHE_DSS_WITH_AES_128_CBC_SHA, SSL_DHE_DSS_WITH_3DES_EDE_CBC_SHA, SSL_RSA_WITH_3DES_EDE_CBC_SHA]

Is at least one of these ciphers accepted by your server?